I fell in the same situation a year ago. I showed my EU driver's license at check in desk but was refused checking in because the clerk stated that driver's license does not show nationality
. Then I gave her my national ID card and everything was ok.
First, the question "please show me your passport" is probablylikely a mantra as stated by @Lassi. Just like when you are checked tickets in a bus but officers ask you to show passes, please
(in the sense of the monthly/seasonal pass used by frequent commuters).
Anyone can get a driver's license in any EU country regardless of his/her nationality. So the driver's license does is surely notcan never be accepted because does not prove EU citizenship. Only the ability to drive a motor vehicle of corresponding EU category.
Now to Schengen agreement: while it is commonly called to be "no borders" (and no checks) between countries, mind that freedom of circulation in Schengen space is a right granted only to Schengen citizens. So you basically have to prove the airline that you are a Schengen citizen to travel the Schengen area.
That is what was told me.